On the uniqueness of convex-ranged probabilities
We provide an alternative proof of a theorem of Marinacci  regarding the equality of two convex-ranged measures. Specifically, we show that, if P and Q are two nonatomic, countably additive probabilities on a measurable space (S, Σ), the condition [∃A∗ ∈ Σ with 0 < P(A∗) < 1 such that P(A∗) = P(B)=⇒ Q(A∗) = Q(B) whenever B∈Σ] is equivalent to the condition [∀A,B ∈ Σ P(A) > P(B)=⇒ Q(A) ≥ Q(B)]. Moreover, either one is equivalent to P = Q.
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